Open Access
2007 Intermittency on catalysts: symmetric exclusion
Jürgen Gärtner, Frank den Hollander, Gregory Maillard
Author Affiliations +
Electron. J. Probab. 12: 516-573 (2007). DOI: 10.1214/EJP.v12-407


We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the influence of a "catalyst". <br> In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument. <br> In G&#228;rtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.


Download Citation

Jürgen Gärtner. Frank den Hollander. Gregory Maillard. "Intermittency on catalysts: symmetric exclusion." Electron. J. Probab. 12 516 - 573, 2007.


Accepted: 1 May 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1129.60061
MathSciNet: MR2299927
Digital Object Identifier: 10.1214/EJP.v12-407

Primary: 60H25 , 82C44
Secondary: 35B40 , 60F10

Keywords: catalytic random medium , Exclusion processes , Intermittency , Lyapunov exponents , Parabolic Anderson model

Vol.12 • 2007
Back to Top